Lemma 3.1 ([1]}) 

Let \(A \subseteq \mathbb {F}_2^n\)  be a linear subspace and let \(\beta > 1\)  be a constant. For any set of polynomials \(p_1, \dots , p_{\beta n} \in I_{d, A}\)  selected uniformly at random, define \(Z = \lbrace  v \in \mathbb {F}_2^n : p_i(v) = 0 \text{ for all } 1 \le i \le \beta n \rbrace \) . Then \(A \subseteq Z\) , and \(\Pr [A = Z] = 1 - 2^{-\Omega (n)}\) .

